Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&, \\ P_{\mathbf{1}}&{}\approx{}&0.008150396943&{}:{}&0.022861266845&{}:{}&0.968988336212&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.007574850487&{}:{}&-0.014551926795&{}:{}&1.022126777282&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.021272205742&{}:{}&0.054080411453&{}:{}&0.924647382804&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.049204237573&{}:{}&-0.150453621409&{}:{}&1.199657858981&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.047223044176&{}:{}&0.122523774417&{}:{}&0.830253181407&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{1}}\)
\(P^-_{\mathbf{1}}\)
\(P^+_{\mathbf{1}}\)
\(Q_{\mathbf{1}}\)
\(I^\prime_{\mathbf{1}}\)
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Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.100871453186&{}:{}&0.528418297529&{}:{}&0.572453155657&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.367664982584&{}:{}&-0.323593937302&{}:{}&0.955928954718&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.015332607192&{}:{}&0.036798257260&{}:{}&0.947869135548&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.029437588678&{}:{}&0.022370615673&{}:{}&0.948191795650&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.007767176080&{}:{}&0.068805035128&{}:{}&0.923427788792&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.074618144197&{}:{}&0.209298432689&{}:{}&0.716083423115&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.016144565582&{}:{}&-0.014209358492&{}:{}&0.998064792911&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.007185176653&{}:{}&0.037639775126&{}:{}&0.969545401527&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.363109318090&{}:{}&-0.697563631723&{}:{}&2.060672949813&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.143397840021&{}:{}&1.143397840021&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.042769422997&{}:{}&0.000000000000&{}:{}&1.042769422997&,\\C^*_{\mathbf{1}}&{}\approx{}&0.246442778780&{}:{}&0.753557221220&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.321045743823&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&2.205989895504&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.091995643151&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.100871453186&{}:{}&0.528418297529&{}:{}&0.572453155657&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.367664982584&{}:{}&-0.323593937302&{}:{}&0.955928954718&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.015332607192&{}:{}&0.036798257260&{}:{}&0.947869135548&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.008150396943&{}:{}&0.022861266845&{}:{}&0.968988336212&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.029437588678&{}:{}&0.022370615673&{}:{}&0.948191795650&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.007767176080&{}:{}&0.068805035128&{}:{}&0.923427788792&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.074618144197&{}:{}&0.209298432689&{}:{}&0.716083423115&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.007574850487&{}:{}&-0.014551926795&{}:{}&1.022126777282&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.016144565582&{}:{}&-0.014209358492&{}:{}&0.998064792911&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.007185176653&{}:{}&0.037639775126&{}:{}&0.969545401527&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.363109318090&{}:{}&-0.697563631723&{}:{}&2.060672949813&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.021272205742&{}:{}&0.054080411453&{}:{}&0.924647382804&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.100871453186&{}:{}&0.528418297529&{}:{}&0.572453155657&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.367664982584&{}:{}&-0.323593937302&{}:{}&0.955928954718&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.015332607192&{}:{}&0.036798257260&{}:{}&0.947869135548&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.029437588678&{}:{}&0.022370615673&{}:{}&0.948191795650&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.007767176080&{}:{}&0.068805035128&{}:{}&0.923427788792&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.074618144197&{}:{}&0.209298432689&{}:{}&0.716083423115&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.049204237573&{}:{}&-0.150453621409&{}:{}&1.199657858981&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.143397840021&{}:{}&1.143397840021&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.042769422997&{}:{}&0.000000000000&{}:{}&1.042769422997&,\\C^*_{\mathbf{1}}&{}\approx{}&0.246442778780&{}:{}&0.753557221220&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.047223044176&{}:{}&0.122523774417&{}:{}&0.830253181407&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.029437588678&{}:{}&0.022370615673&{}:{}&0.948191795650&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.007767176080&{}:{}&0.068805035128&{}:{}&0.923427788792&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.074618144197&{}:{}&0.209298432689&{}:{}&0.716083423115&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.143397840021&{}:{}&1.143397840021&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.042769422997&{}:{}&0.000000000000&{}:{}&1.042769422997&,\\C^*_{\mathbf{1}}&{}\approx{}&0.246442778780&{}:{}&0.753557221220&{}:{}&0.000000000000&. \end{alignedat} \]
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